Recursive Intergenerational Utility in Global Climate Risk Modeling

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(1)99s-40. Recursive Intergenerational Utility in Global Climate Risk Modeling Minh Ha-Duong, Nicolas Treich. Série Scientifique Scientific Series. Montréal Novembre 1999.

(2) CIRANO Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Le financement de son infrastructure et de ses activités de recherche provient des cotisations de ses organisationsmembres, d=une subvention d=infrastructure du ministère de la Recherche, de la Science et de la Technologie, de même que des subventions et mandats obtenus par ses équipes de recherche. CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and research activities are funded through fees paid by member organizations, an infrastructure grant from the Ministère de la Recherche, de la Science et de la Technologie, and grants and research mandates obtained by its research teams. Les organisations-partenaires / The Partner Organizations $École des Hautes Études Commerciales $École Polytechnique $Université Concordia $Université de Montréal $Université du Québec à Montréal $Université Laval $Université McGill $MEQ $MRST $Alcan Aluminium Ltée $Banque Nationale du Canada $Banque Royale du Canada $Bell Québec $Développement des ressources humaines Canada (DRHC) $Egis $Fédération des caisses populaires Desjardins de Montréal et de l=Ouest-du-Québec $Hydro-Québec $Imasco $Industrie Canada $Microcell Labs inc. $Raymond Chabot Grant Thornton $Téléglobe Canada $Ville de Montréal © 1999 Minh Ha-Duong et Nicolas Treich. Tous droits réservés. All rights reserved. Reproduction partielle permise avec citation du document source, incluant la notice ©. Short sections may be quoted without explicit permission, provided that full credit, including © notice, is given to the source. Ce document est publié dans l=intention de rendre accessibles les résultats préliminaires de la recherche effectuée au CIRANO, afin de susciter des échanges et des suggestions. Les idées et les opinions émises sont sous l=unique responsabilité des auteurs, et ne représentent pas nécessairement les positions du CIRANO ou de ses partenaires. This paper presents preliminary research carried out at CIRANO and aims at encouraging discussion and comment. The observations and viewpoints expressed are the sole responsibility of the authors. They do not necessarily represent positions of CIRANO or its partners. ISSN 1198-8177.

(3) Recursive Intergenerational Utility in Global Climate Risk Modeling* Minh Ha-DuongH, Nicolas TreichI Résumé / Abstract Ce texte étudie la différence entre l'aversion relative au risque et la résistance à la substitution intertemporelle dans la modélisation du risque climatique. Les préférences récursives stochastiques sont utilisées dans un modèle numérique stylisé utilisant les scénarios préliminaires GIEC 1998 sur l'économie et le climat. On montre qu'une aversion au risque plus forte conduit à augmenter le niveau optimal de taxation de l'énergie. Augmenter la résistance à la substitution intertemporelle a le même effet qu'augmenter le taux d'actualisation, tant que le risque n'est pas trop grand. Nous discutons les implications de ces résultats pour le débat sur l'actualisation et la durabilité sous incertitude. This paper distinguishes relative risk aversion and resistance to intertemporal substitution in climate risk modeling. Stochastic recursive preferences are introduced in a stylized numeric climate-economy model using preliminary IPCC 1998 scenarios. It shows that higher risk aversion increases the optimal carbon tax. Higher resistance to intertemporal substitution alone has the same effect as increasing the discount rate, provided that the risk is not too large. We discuss implications of these findings for the debate upon discounting and sustainability under uncertainty.. Mots Clés :. Utilité récursive, risque, actualisation, durabilité, climat. Keywords:. Recursive utility, risk, discounting, sustainability, climate. JEL : Q20, D81, D90, H43. *. Corresponding Author: Nicolas Treich, CIRANO, 2020 University Street, 25th floor, Montréal, Qc, Canada H3A 2A5 Tel.: (514) 985-4000 Fax: (514) 985-4039 email: [email protected] We thank Christian Grollier, Jean-Charles Hourcade, Alan Manne, Anne Épaulard, Aude Pommeret and seminar participants at GREEN (Québec), AFSE (Toulouse), CIRED (Paris), EAERE (Oslo) and Ateliers ColumbiaCIRANO (Montréal) for useful discussions, while remaining errors are only ours.. †. CIRED-CNRS Paris. ‡. LEERNA-INRA Toulouse and CIRANO.

(4) 4. HA DUONG AND TREICH. Non-technical abstract In this paper, we propose a modeling framework to analyse risky and intergenerational issues. We then apply it to the issue of climate change. This framework is stochastic recursive utility. It allows to represent a broader range of preferences than what is usually found in the litterature. The structure of the paper is the following. The one-page introduction frame the issue. It requires a graduate level in economic theory to be read. Section two presents an ultra-long period model of the climate risk named ULTRAL. Section three presents the calibration of the model, where we discuss numeric values. Section four presents and discuss the results. Section five concludes. The appendix defines the model and data used. Lastly, there is the list of the 35 references. The tool used in this paper, stochastic recursive utility, appeared in economic theory about twenty years ago. To our knowledge, it has not been previously used in integrated assessment modeling. On the theoretical side, it allow to put in perspective the debate on discounting under uncertainty. Policy interpretations of the results should be conducted with extreme care since this more a methodological paper. The simplest insight is that the assumption of a positive expected climate damage is not needed to induce some action now. In this model, climate change is beneficial nine times out of ten, nevertheless there is some precautionary abatement. In a sequential decision making process, the initial level of abatement was found to be sensitive the date of policy revision. The earlier is policy revision scheduled, the lower is initial carbon tax level. The model also show that larger risk aversion strengthens optimal pollution control and that previous works possibly underestimated the sensitivity of their results to the shape of the utility function..

(5) RECURSIVE INTERGENERATIONAL UTILITY. 5. 1. INTRODUCTION This paper is motivated by two important economic aspects of climate change economics: intergenerational equity and risk. A classic way to introduce risk is to consider an utility function with constant relative risk aversion , such as:.

(6) . (1). A classic way to introduce the issue of intergenerational equity is to consider an intertemporal objective function such as:.

(7) "# %$ &

(8) ' )( (2)

(9)

(10)

(11) ! (+*#, where represents population and consumption levels, see [2, 19, 30]. This paper focuses on and on the two parameters of the intertemporal objective $ function function: , the discount rate for the second time period, and - , the resistance to intertemporal substitution. The simplest possible model known to integrate these three parameters may be stochastic recursive utility, initially introduced by Kreps and Porteus [18]. To our knowledge, all previously published models of the climate risk literature use simpler utility functions restricted to - . [24, 20, 16]. This situation has left open a set of question we propose to examine. From a theoretical point of view, let us first remind that resistance to substitution - controls the attitude toward variations in consumption across time, under certainty. On the other hand, risk aversion controls the attitude toward variations in consumption across states of the world, at a given date. So the restriction - / is suspicious since it links two conceptually different aspects of preferences. We will examine whether there is empirical evidence and economic rationale to support this constraint, or whether it is an unfortunate sequel of expected utility models. From a technical point of view, this restriction is particularly inadequate when analyzing the question of equity under uncertainty. It prevents from considering plausible situations where society has high risk aversion together with low resistance to substitution, or vice versa. Another direct consequence is that the effect of resistance to substitution and risk aversion cannot be conveniently investigated: A change in the curvature in the utility function does not permit an unambiguous interpretation since the resistance to substitution and risk aversion move together. We will examine the effect of risk aversion on optimal climate policy and the effect of resistance to substitution on sustainability. Section two presents an ultra-long period model named ULTRAL. Section three presents the calibration of the model, where we discuss numeric values for - and . Section four presents the results, where we study the sensitivity of optimal control strategies to specifications of the utility function, to uncertainty, and to a sustainability constraint. Section five concludes..

(12) 6. HA DUONG AND TREICH. 2. MODEL The ULTRAL model is a stylized ultra-long period classical macroeconomic model, 021 stochastic with a recursive utility function, where climate change due to accumulation affects productivity and the efficiency of investment. The time step is 30 years. There are four dates 3 starting from 1990, therefore three periods representing current, next and future generations. There are two states of the world, 4 and 5 , referred to by the index 6 . The model is coded in GAMS and solved using nonlinear optimization with MINOS or CONOPT. In this section, we will discuss the equations of ULTRAL. For convenience, they are presented all together in the annex. Equations can be divided into three blocks. The first block A.1–A.5 describes classical economic 021 growth with a natural resource [33]. The second block A.6–A.9 computes accumulation, concentration, temperature increase and climatic damages. The third block A.10–A.11 describes preferences. 2.1. Climate-economy The macroeconomic block is built around a capital-labor-energy production function. Population 87 is exogenous. All variables such as capital 9;:< 7 , energy input =>:< 7 and production ?:< 7 are per-capita. Technical progress @ 7 is exogenous. Productivity decreases with global warming through a multiplier AB:< 7 . Capital depreciates and is renewed according to a fixed saving rate C 7 as described in equation A.4. The control variable DE:< 7 is a multiplier to the cost of the energy input. Exante, the reference cost of energy is F 7 . The amount of energy input =>:G< 7 entering the production function depends upon F 7 DE:< 7 its ex-post cost. Broadly speaking, H is the percentage of energy tax over the reference price. It is constrained DE:G< 7 to be positive. The model runs a cost-benefit analysis. The cost of control is computed as follows. At period 3 , capital 9 :G< 7 is fixed from the previous period’s investment. Productivity @ 7 and population 7 are exogenous. Equations A.3 (discussed below) 0 and A.1 together determine production ? :< 7 and energy = :G< 7 . Consumption :< 7 and investment follow since the saving rate is fixed. Economic activity decreases with DI:< 7 . Solving ?:< 7 and =>:< 7 equations yields:. 7 F 7 'L +( *N( M A :G(+< 7 *N( M 9 :G< 7&J 9

(13) PO D :(Q< *N7 *NM M (3) (+*NM 7 AB:< 7 J F 7+ ( 9;:< 7 J 9

(14) PO D :MR< 7 *S( ( (4) (Q*NM (+*NM where superscripted parameters ?

(15) and =

(16) denote ex-ante first period values, and where 9

(17) is fixed. Note that for values of T much smaller than unity, production than energy =V:G< 7 to changes in energy cost DE:< 7 . Assuming ?U:< 7 is much less elastic V . for example 3 , A

(18) <:. and TXW in the above equations, then energy. @ ? :< &7 J ?

(19) K. @ =>:< 7 J =

(20) K.

(21) 7. RECURSIVE INTERGENERATIONAL UTILITY. TABLE 1.. YVZ+[ \^]E_ `ba Z+[ \ `bc Z+[ \. Reduction cost curve in first period. 0. 10%. 50%. 100%. 200%. 0 0. -9% -0.4%. -24% -1.7%. -51% -2.8%. -68% -4.5%. consumption =

(22) < : is inversely proportional to the tax D

(23) < : , while output ?

(24) < : only 'L decreases as D

(25) < : . The corresponding reduction cost curve is numerically shown in Table 1. Equation A.3 stems implicitly from a behavioral assumption: the energy sector maximizes its profit blindly, in a sequence of static equilibria. Due to this assumption, the model can be seen as a two-stage optimization structure: economic agents optimize within a 30-years horizon, and only the central planner has an intergenerational point of view. Given that the price of output is 1 and the price of energy is F 7 D :G< 7 , this assumption of intragenerational agent optimization implies a well-known Cobb-Douglas function property, that:. ?U:< 7. T F 7 D :< 7 = :< 7 This corresponds to equation A.3. The share of energy T disappear because the equation at date 3 is scaled with the equation at date 1. Note that equation A.1 is also scaled this way. Adjustment costs kick-in to decrease consumption when D :< 7 increases faster than 2% per year, that is 1.81 per period. This is equation A.5. Finally, the model represents the sequentiality of the decision making process for climate policies. It is known from the start that the policy computed initially will be revised at date 3 edgf . This is why all variables are indexed by the state of the world 6 and there is a constraint DEhi< 7 DEj'< 7 for 3lk d f . This is a stochastic dynamic programming technique which would not be needed if the concept of “time unfolding on a stochastic tree” was built in the modeling languages. This sophistication will only be important when we analyze uncertainty in the last subsection 4.3, since before we will assume d f nm . The benefits of reducing emissions are computed as follows. The carbon cycle central variable is a simple non-controversial index used by scientists and policymakers: global cumulative fossil and industrial carbon emissions since 1990. Let o 7 :< be this index computed by equation A.6 in GtC. ULTRAL uses a black-box linear representation based02 upon 1 seven major models concentration levels described in [8]. At each period, these models associate p 7 o :< with :G< 7 . A linear relationship was estimated using least-square o interpolation p to determine qB7r and q 7

(26) in equation A.7. Equation A.6 computes :< 7 from :< 7 using these linear coefficients..

(27) 8. HA DUONG AND TREICH. Temperature increase sI:G< 7 is estimated in equation A.8 using a 2.5-degrees increase for a doubling of pre-industrial 021 concentration. The 275 ppm number represents pre-industrial levels of in the atmosphere. Equation A.9 represents the economic effects of climate change At:G< 7 as an overall mean-preserving productivity jump. It can be interpreted as an extreme irreversible event, occurring at date dgu , which changes the global climatic conditions in such a way that a major part of the capital and technological stock has to be replaced. To account for the possibility of adaptation and published impact estimates [22], the expected value of the productivity jump is zero: before d u , the model sets A 7 . At a given4 date d u , productivity drops permanently a lot with w a small v (state probability v (state ), or increases a little with a large probability 5 ). The magnitude of the fall depends quadratically upon temperature change. 2.2. Recursive utility We now describe the representation of preferences with recursive utility. The motivation is to separate attitude towards intertemporal substitution from attitude towards risk. Koopmans presented axiomatic deduction of recursive preferences [17]. 7 an7Gy early He wrote x 7. x

(28) and called the aggregator function. Kreps and Porteus [18] generalized this structure to stochastic models:. x 7z 7 R { 7 x | G7 y

(29) & where { 7 is the certainty equivalent of future utility evaluated with information available at date 3 . Let it be computed using } R~ , a von Neuman-Morgenstern utility function:. { 7 g| } g

(30) & 7 } '| . This class of preferences separates risk aversion — tied to the certainty equivalence function { or } — from intertemporal substitution — tied to the aggregator function . Interestingly, they satisfy the properties of intertemporal consistency and stationarity of preferences. The specification of and } is not unique. Define & V } }

(31) &&. . |. and x 7R~> } x 7&~ . Then x 7i 7 7 x 7Gy

(32) define the same preferences as before. Here we follow Epstein and Zin [10] and take and { 7 isoelastic:. K

(33)

(34) (+*#( , q ! { 7 g| K&7 |

(35) U (Q*( . (5) (6).

(36) RECURSIVE INTERGENERATIONAL UTILITY. 9. Three positive parameters , , and - characterize the preferences. The - in the aggregator function is the constant resistance to intertemporal substitution. The parameter is the Arrow-Pratt constant relative risk aversion. The is next period’s felicity discount factor. Since it is known that isoelastic functions tend continuously toward the logarithm when - or tend towards unity, we will simply ignore the cases -. or . It is possible to choose q according to the units of consumption and utility. In one equation, the recursive preferences in the model are:. > R 7

(37) '

(38) U (Q*#, x 7

(39) ' (7) !) & 7 x | 7Gy

(40) (+* ~ l As usual in the literature, q is set to . In order to explain the assumptions underlying this equality, let be the expected growth of utility beyond the last | . ! x . Then equation 7 is equivalent to: period, that is xz g

(41) UN (+*#, q J x

(42) . (8) (Q* ! 2 The equality q. arises from the two following conditions: x X . together with . The first condition is numerically important. Our experience shows that most other formulations for q lead to non-convergence when solving for various , and - . The second condition is a first approximation potentially leading to mathematical problems with a zero utility discount rate, as . implies q . Divergence under assumptions of zero growth and zero discounting is normal since recursive preferences generalizes classical intertemporal utility. So, the problem disappears if either some growth is assumed or,as here, utility discount rate never goes to zero. As shown equation A.10, the certainty equivalent of next generation utility { 7 x | 7Gy

(43) is explicitly represented by the state variable 7 defined by equation A.11. In that equation, the 3 m case is the final condition introduced above, where consumption level is not computed explicitly, but approximated by U , where 9; is itself taken from equation A.4. 8 7 Finally, the guideline to introduce population was to retrieve, in the special case - / , the standard intertemporal expected utility model:. 87 x 7 87R 7

(44) ' ! 7.87Gy G7

(45) y ' +( *#( ,.

(46) 10. HA DUONG AND TREICH. 3. CALIBRATION 3.1. Preferences We now turn to the calibration of stochastic recursive utility functions. The utility discount rate is set at 1% per year in the base case. Values 0, 0.5 and 2% per year will also be discussed. This range of values is typical in the literature, see for example Portney and Weyant [30]. The other two parameters - and need more discussion. First of all, the golden rule of optimal economic growth gives a theoretical way $ for estimating - , given , , and real risk-free marginal productivity of capital ¡ . Let it be quickly reminded. Assuming in equation 2 that ¢ ! &

(47) and £ ! R

(48) , and that , and $ are small, the marginal rate of substitution between future and present consumption is:. ¤P¥V¦ § <

(49) X $ ! - (9)

(50) < W ! ¤¨¥V¦ ©§ $i ! -. It follows from the first optimality condition ! ¡2.

(51) that ¡2. $ Decreasing the annual utility discount rate can be compensated with a higher - . $ With n , given in the 1 to 2 percent per year range, annual global population $ growth between 1 and 1.5, and ¡ in between 3 and 6 percent per year, this gives - in the 2 to 7.5 range. But these calculations are subject to controversies. For example, Cline [7] considers a wedge of market distortion reducing ¡ by 50%,. whereas Nordhaus [26] states that he does not see how anyone could convince himself that ¡ is below 6%. Following the seminal Kreps and Porteus’ paper [18], another interesting thought experiment is to compare utility if 6 uncertainty is resolved at period 3 to utility if 6ª6 uncertainty is resolved at period 2:. 6 6ª6 . x

(52)

(53) ' q

(54)

(55) ! q

(56) ! x

(57)

(58) ' q

(59)

(60) !);¬ g q

(61) ' ) ! . «

(62) U +((+**#, «

(63) (+(Q**#, (+Q( **#,. It can be shown that early information 6Q6 is preferred if and only if )® - . In this sense, assuming that society exhibit a taste for early information (or a distaste for late information) would limit the range of plausible values for - and . The usual intertemporally additive expected utility model, corresponding to - e , is equivalent to assuming indifference to the timing of uncertainty resolution. Research on saving and portfolio behavior analysis derived empirical estimates of and - directly. In a seminal paper, Hall [14] estimated the response of saving to the real interest rate, that is the elasticity of intertemporal substitution ( J - ). All estimates presented by Hall report small elasticity of substitution that may well be zero. Ogaki and Reinhart [28] argued that Hall’s model was misspecified and found estimates of resistance to substitution - that vary between 2.2 and 3.1..

(64) RECURSIVE INTERGENERATIONAL UTILITY. 11. Another important issue with current - and parametrisation is the equity premium puzzle as exposed by Mehra and Prescott [21]. Mehra-Prescott and Hall models have been criticized on the ground that they fail to separate resistance to substitution from risk aversion. In doing so, Epstein and Zin [11] found estimates for - that vary between 1.2 and 5 and estimates for that vary between 0.8 and 1.3. Using a similar approach, Normandin and Saint-Amour [27] estimates for and are in the range of 1.5. It is also noticeable that they accepted the hypothesis - n while Epstein-Zin rejected the expected utility model. In an experimental framework, Barsky, Juster, Kimball and Shapiro [4] reported estimates of each parameter based on individual responses to hypothetical situations. They found high heterogeneity among individuals. A striking results of their study is that risk aversion and resistance to substitution are not correlated at all across individuals. We found no decisive empirical evidence either to reject or to accept intertemporal expected models (- ¯ ) nor to choose some specific values for - and . Arrow [1] reports that 1.5 is the currently available best-guess for both parameters. ULTRAL uses values in the 0.6 to 4 interval, with emphasis upon 1.3 and 2.0. 3.2. Socio-economic scenarios The socio-economic parameters are calibrated using preliminary Intergovernmental Panel on Climate Change 1998 Scenarios, see the annex. There are four scenarios A1, A2, B1 and B2. Each scenario proposes a particular future. It is important to stress that these scenarios are non-mitigation scenarios with respect to climate change. According to the IPCC data distribution center,scenario A1 represents a world of very rapid economic growth, low population growth and rapid introduction of new and more efficient technology. In this world, people pursue personal wealth rather than environmental quality. Scenario A2 describes a world with high population growth with less concern for rapid economic development. Scenario B1 accounts for dematerialization and for the introduction of clean technologies. Scenario B2 describes a world in which the emphasis is on local solutions to economic, social, and environmental sustainability. Parameters displayed in the annex correspond to the A1 scenario. Space constraints prevent us from reporting the four parameters set, but all numeric simulations were conducted on all four scenarios. As can be seen on Figure 1 panel a, future carbon emissions profiles span a large interval. Population 87 and carbon content of energy ° 7 are drawn from IPCC scenarios. Production and capital parameters are based upon order of magnitudes usually accepted in theoretical and empirical literature. The saving ratio is C 7l ±²~³#´ for all 3 . In the production function, the share of labor is ²~ µ , and the share of energy is about ten times smaller than for capital, that is ¶ ·^~ ¸Nµ and T ·^~ #m . The reduction cost curve in period 1 implicit to this is shown in Table 1..

(65) 12. HA DUONG AND TREICH. . The ex-ante levels of production ?

(66) and energy =

(67) correspond to DE:G<

(68) . and are given parameters. The initial level of capital 9

(69) is set at three times ?

(70) . Finally, capital depreciation factor ¹ and investment accumulation º are based upon a 5% annual depreciation rate. Calibration of @ 7 and F 7 was done as follows. Control DE:< 7 and damage AB:< 7 were set to unity, i.e. ignored. This allowed to compute values for baseline energy costs F 7 by using scenario values for ? :< 7 and = :G< 7 in equation A.3. Then coefficients for technical progress @ 7 were computed by solving equations A.3 and A.1 recursively. The effect of this calibration is that when there is no climate damage, growth tracks the mark-up scenario from IPCC. There is a 5% probability that one degree of warming causes a 10% drop in productivity. That can be related to the EMF 14 uncertainty subgroup model comparison framed in [25]. That note suggested, with a 5% probability, to multiply the damage function by 7.8. In this framework the 10% drop is equivalent to a 1.28% unitary damage coefficient, which is in line with values used in the literature [5]. In the model, adding 3.5 degrees to the climate temperature sensitivity would have the same effect as multiplying economic damage sensitivity by a factor of about 5.8. 3.3. Baseline optimal control ~ ¸ » ~ ¸ , As a reference optimal control scenario, we consider the -. utility function. Policy revision and climate event occurs at the same period d f. dgu em , for zero expected climate damage. 021 Panel b of Figure 1 displays the optimal emissions levels with ULTRAL in each of the four IPCC markup scenarios. For each scenario, emissions rise very slowly during first period, until 2020. Then the emissions of each scenario strongly diverge. In scenario B1 for instance, there is a stabilization of emissions around 8 GtC, while in scenario A2 emissions increase sharply from 5 GtC in 2020 to 16 GtC in 2050. These results bear no implications for the timing debate, given the coarse 30-year time step. The panel c of the same Figure 1 presents another view of the same results, where the vertical axis shows optimal control level D :< 7 , i.e. the energy tax, instead of carbon emissions. The horizontal time scale goes only to 2050, since D < : equals D « < : by assumption. Both the carbon tax and the overall level of emissions are higher in scenario A2 than in any other scenarios in 1990 and 2020. Note however that this is no longer true in 2050. There is higher control in 2050 for scenario A1 than for scenario A2. This can be related to the growth effect: per capita production is very high in scenario A1, the economy expands at an average rate of about three percent to 2080. As a consequence, investment in control will be relatively less costly in the long run in scenario A1..

(71) 13. RECURSIVE INTERGENERATIONAL UTILITY. ¼¾½ _¿ À. ÁÂ½ _¿ À. FIG. 1. ULTRAL reference profiles. Panel a Baseline carbon emissions according to IPCC preliminary 1998 scenarios. Panel b Optimal carbon emissions with and utility function. Panel c Corresponding optimal control. Y2\GÃ [ Ä. ctEtLt (GtC). A2. 20. A1. 15. a. B2 10 B1 5 1990. 2020. 2050. 2080+ t. ctEtLt. A2. (GtC) 20 b 15. A1 B2. 10 B1 5 1990. 2020. 2050. 2080+ t. Xt. 2 A2. c. A1 1.5. B2 1 B1 t 1990. 2020. 2050 +.

(72) 14. HA DUONG AND TREICH. ¼ ½ _¿ À i. FIG. 2. Optimal control (energy cost multiplier constant .. Y Ä RÃ Å^Æ ) for varying relative risk aversion Á. , given. Xt 3 4.0. 2.5. 2.6 2.0. 2. 1.3. 1.5. 0.8 0.5. 1 t 1990. 2020. 2050+. 4. RESULTS 4.1. Sensitivity to utility function coefficients In this section, we examine the sensitivity of the optimal control profile to the three parameters of the social planner’s preferences , - and . What is the effect of risk aversion on the optimal climate policy? Some studies [6] found that risk aversion warrants aggressive abatement action while another [24] did not find a large effect of risk aversion in his sensitivity analysis. A problem is that these results are ambiguous since they are based upon a change in the curvature of the utility function representing both resistance to substitution and risk aversion. The isolated effect of risk aversion was first investigated in the literature on savings under uncertainty. Selden [31] and Weil [34] used recursive preferences to show that optimal precautionary savings tends to be determined solely by the elasticity of intertemporal substitution and not by the risk aversion parameter. Similar results were also found in the literature of dynamic resource management [15, 9]. Figure 2 displays the effect of relative risk aversion on the optimal climate policy in scenario A2. We consider varying from 0.5 to 4. Relative risk aversion leads unambiguously to an increase in the carbon tax in order to face less risk in the future. For example, increasing from 1.3 to 2.6 leads to increase D

(73) from 1.29 to 1.65. Doubling risk-aversion does more than doubling the optimal energy tax level..

(74) RECURSIVE INTERGENERATIONAL UTILITY. Á ½ _¿ À w. FIG. 3. Optimal control (energy cost multiplier substitution , given constant . Xt. ¼. 15. Y Ä +Ã Å^Æ ) for varying resistance to intertemporal. 2.5 2.25 2 1.75. 0.5 0.8. 1.5 4.0 1.25. 1.3. 2.6 2.0. 1 t 1990. 2020. 2050+. With ULTRAL, the result that risk aversion increases optimal control is robust with respect to the baseline scenario and the value of - . Turning to the effect of resistance to intertemporal substitution, Figure 3 displays the optimal carbon tax for different values of - . Numerical simulations show that increasing resistance to substitution tends to rotate the optimal control path counterclockwise, towards less control in the current period and more control in the future periods. This goes in the direction of reducing the differences of consumption levels between generations. The intuition explaining this results is therefore that more emphasis on equity — larger - — imply higher current emissions levels to smooth the differences with following, richer generations. This is a discounting effect. Since control costs are borne in the short term and benefits will be received in the long term, increasing - implies less control in the current decade. Indeed, as equation 9 shows, assuming a positive rate of growth, an increase in $ - increases the cost discount rate. To some extend, and - have the same effect. ¢. e . ³. For example, for and , the optimal tax rate in first period is 15%. Model runs show that lowering the discount rate to 0.5 has basically the same effect as lowering - from 2 to 1.3. Sensitivity of results to joint variations in - , and are reported in table 2 ~¸ for the B2 scenario. Comparing the - ¢±. cases with - ¢± ³ , the optimal tax level does not change much: from 15% to 17% in one case, from 77%.

(75) 16. HA DUONG AND TREICH. TABLE 2.. Sensitivity of the optimal tax level at. ¼. Resistance to substitution ( ) Aversion to risk ( ) Annual discount rate (gives ). Á. É Y E ] _) Ã R Ê Æ Optimal tax level ( Æ. Ç^½È. ¼ Á. to the utility function parameters ,. É. and .. 2 2 1%. 2 1.3 1%. 1.3 2 1%. 1.3 1.3 1%. 2 2 0.5%. 2 1.3 0.5%. 1.3 2 0.5%. 1.3 1.3 0.5%. 15%. 0%. 72%. 17%. 77%. 18%. 83%. 80%. to 80% in the other. This seemingly small sensitivity disappears when only - or change. This illustrates that parameters - and tend to cancel out when they move together. In the light of previous results, the explanation is straightforward. Increasing risk aversion leads to increase early control to prevent the climatic catastrophe. But increasing resistance to substitution in the same time leads to decrease control to favor the current generation which is the poorest in average. Thus the use of intertemporally additive expected utility model (- Ë ) underestimates the sensitivity of the results since the two parameters move together and triggers these two opposite effects. Three main results emerged from this section. First, increasing risk aversion has a positive effect on the control rate. Second, increasing resistance to in$ tertemporal substitution - has the same effect as increasing the discount rate , it decreases the initial level of control. Third and consequently, the classical expected utility model induces a bias toward a more neutral climate policy. 4.2. Discounting and sustainability under uncertainty $ If - and have the same effect, why use two parameters? The second result above suggests that there is some redundancy in the parametrisation of recursive $ utility function. One possibility is to eliminate time preference and go for a resistance to substitution higher than the usual -. log-of-consumption model, for example - Ì³ . A more widespread attitude is to restrict utility functions to $ the - Í subset and focus the discussion on . $ In this section, we first explore with ULTRAL the difference between - and when it comes to the question of sustainability. The discussion of these results will allow to show limitations of the attitude focusing only on the discount rate. There are many definitions of sustainability. Amongst them, let us single out Solow’s [32] formalization as non-decreasing utility through time. This interpretation imposes a chain of obligation between each successive generation which can very conveniently be represented with recursive utility. Sustainability under certainty is defined x 7 k±x 7Gy

(76) , for all 3 . A natural generalization under uncertainty discussed theoretically in [3] is:. x 7 k { 7 x | G7 y

(77) . (10).

(78) RECURSIVE INTERGENERATIONAL UTILITY. ¼ ½ _¿ À l. Á ½ _¿ À w. FIG. 4. Optimal control (energy cost multiplier and . Xt. 17. Y Ä RÃ Å^Æ ) with the sustainability constraint, for. 2.5 2.25 A2. 2 1.75 1.5. B2. 1.25. A1. 1. B1 t 1990. 2020. 2050+. By using a certainty equivalent function { 7 &~ , this definition allows welfare to decrease if bad luck arises, but to an extent controlled by the risk aversion parameter. Pezzey [29] suggested to consider sustainability as a prior constraint on the optimization program. It can also be used as a criteria to evaluate ex-post optimal paths. We conducted both analysis. We examined, for 76 model runs using varying parameters sets, which ones were sustainable in the above sense. As could be expected, scenario A2 was the least often sustainable, followed by B2, A1 and B1 in this order. This explains why, when solving the model under the sustainability constraint 10, in the model x 7 k» 7 , the effect is important only for A2 and B2 scenarios (compare figure 4 with the unconstrained optima figure 1 panel c). Discounting felicity at a high rate, two or three percent per year, led to un$ sustainable pathways with two interesting exceptions. For A1 and Î³NÏ , a ~ ¸ did not. For B1, risk aversion of n ³ restored sustainability where . ¢. ³ a resistance to substitution led to sustainable optima. The latter result is ~¸ $ interesting since -. did not lead to sustainability. Here and - have opposite consequences on the optimal path. Our explanation is that substitution effects depend upon the absolute level of welfare, whereas discounting depends upon time. As long as there is a monotonous relationship between time and wealth, i.e. growth, substitution and discounting are broadly equivalent. But the stochastic model used here breaks this relationship for $ some states of the world. This is why and - have opposite effects..

(79) 18. HA DUONG AND TREICH. This result illustrates that, from the sustainability point of view, there is more in resistance to substitution than simply in discounting. Consequently, focusing the $ debate on intergenerational equity only on is problematic. This adds to other problems with the discounting approach when analyzing very large scale and long term projects. There is an endogeneity question since large scale projects affect the growth path of the economy. This path should not be considered as exogeneous. There is also the need to introduce risk when considering long-run projects. In their recent path-breaking papers, Weitzman [35] and Gollier [12] examine directly how macroeconomic uncertainty affects the choice of the discount rate. They prove that it reduces significantly the discount rate for very long-term projects, maybe at its lowest possible value. On the other hand, usual practice is to increase the discount rate for risky project. For anthropic climate change, the correlation of risks between emissions, economic activity and impacts therefore lead to ambiguity. We do not see how analysis of sustainability could avoid to deal with nonmarginal differences in the welfare levels. More general models of intergenerational utility, such as the stochastic recursive framework, seems to us more appropriate. It appears that increasing the discount rate is not the same as increasing the resistance to intertemporal substitution. 4.3. Sensitivity to uncertainty In this section we present the sensitivity of optimal control to uncertainty in ULTRAL. Results up to this point assumed d f /dgu) em , in other words that bifurcations occurs after 2050. We examine now how an earlier climatic event affects the optimal control. Still assume d f ±dgu , that policy is revised in the same period as climate change happens. Figure 5 illustrates optimal strategies in a sequential framework. In a sequential framework, the central result is near-term policy: How does the possibility of sooner climate event affect the optimal first-period control? Results show that an later settlement of climate change uncertainties implies a higher control in the short run. This result, which Ha-Duong, Grubb and Hourcade already found in [23], calls some discussion. At first, the potential for earlier climatic event implies larger total lifetime impacts since they come sooner and thus last longer. This could lead to increase control in the current period. But early damage also allows for a better adaptation strategy since efforts may be developed sooner. There is thus also a substitution effect between ex post and ex ante control. Figure 5 shows that the substitution effect seems to dominate. To confirm this hypothesis, we examined the difference between D

(80) sensitivity to dgu and its sensitivity to d f . This needs some theoretical comments first..

(81) RECURSIVE INTERGENERATIONAL UTILITY. Y Å^Ã Æ [ Ä. ¼l½ _¿ À. 19. FIG. 5. Optimal control strategies for different period of uncertainty resolution, assuming that policy revision and climatic event coincide, for and .. Áw½ÐÈ. Xt 3.5 TE=TR=4 TE=TR=3. 3. TE=TR=2 2.5. 2. 1.5. 1 t 1990. 2020. 2050+. In many situations, assuming d u »d'f is realistic. In some financial markets for example, the movements of exchange rates (d'u ) are observed in real time by operators which adjust their portfolio within minutes (d f ). For climate change, this is different. There is ongoing climatic research and it is hopeful that scientists and policymakers will know well in advance the severity of climate change. This suggests that precautionary action is possible, and gives background to the assumption d f k dgu of early perfect foresight. On the other hand, ULTRAL does not exclude either the opposite situation d f ®dgu . This is a way to introduce inertia into the model. International decision procedures usually involves a large time lag. Another source of time lag is the natural variability of climate which complicate detection and attribution of anthropic climate change. This is why in ULTRAL there is no a priori relation between d u , the date of climatic event, and dgf , the date of policy revision. Ñ Table 3 summarizes sensitivity of first-period optimal control D

(82) to the un ~ ¸ in the A2 scenario. certainty parameters dgu , d f and risk aversion , with -. Results are very sensitive to d f indeed. Compare the line d f nm with d f e¸ . There is much lower initial control with earlier policy revision. For the lower values of risk aversion k ³ , earlier learning may even lead to no effort at all. This remains when comparing d f /m with d f ³ . This a standard wait and see effect which can be linked to the theory of irreversibility. The prospect to be able to adapt policies sooner gives an incentive to differ control..

(83) 20. HA DUONG AND TREICH. TABLE 3.. Ó Ô ½ÈÀ ÓSÔE½ ÓSÔE½Ð×. YÒZ Ã+Å^Æ ];_ ) to uncertainty parameters ÓÔ , ÓÕ Sensitivity of optimal initial tax ( ÁV½ À ¿ Ö À ÓSÕÂ½ÂÀ × ¿ Ö ÓÕÂ½ÐÈ ÓSÕÂ½ ÓSÕ;½Ð× Áw½ _¿ À Áw½ 24% 01% 55%. 27% 11% 74%. 28% 10% 81%. 0 0 29%. 28% 10% 81%. Á. and .. Áw½ÐØ ¿ Ù 48% 65% 81%. Results confirm the hypothesis that the substitution effect dominates the damage effect. Table 3 shows that optimal control is much less sensitive to d u than to dgf . Chancey Starr has a saying he calls the Noah principle of decision under uncertainty: “Predicting rains does not count, building ark does”. Studying the sensitivity of results to uncertainty parameters with recursive utility is all the more interesting that it most published work to date has been led with a logarithmic intertemporal expected utility function [20, 24, 16]. Yet Gollier, Julien and Treich [13] have shown that the logarithmic case is precisely the limit case where better information has no effect on the level of current consumption in in their proposition 1). With respect to this a stock pollutant model (take Ú. initial motivation of our work, numerical simulations led with parameters close to 1 were non conclusive..

(84) RECURSIVE INTERGENERATIONAL UTILITY. 21. 5. CONCLUSION In this paper, we considered the generalized Kreps-Porteus stochastic recursive preferences that permit to cut the link between risk aversion and resistance to intertemporal substitution - . They were used to build a stylized intergenerational integrated assessment model, ULTRAL. The model has zero climate damages on average. This highlight a simple insight: the assumption of a positive expected climate damage is absolutely not needed to induce some precautionary abatement in the optimal path. Another characteristic of the model is the distinction between the date of policy revision d f and the date of effective climate change dgu . It was found that the initial level of abatement is much more sensitive to the former than to the latter: The earlier is policy revision scheduled, the lower is initial carbon tax level. On the theoretical side, stochastic recursive utility contributes directly to the discussion on discounting in cost-benefit analysis [2, 19, 30] by providing an unifying framework for the main different points of view. For example, many climate change models use the discounted sum of logarithmic expected utility, corresponding to the - Í. case. In this paper, we analyzed the sensitivity of the optimal energy consumption path to the specifications of the utility function. The first result is that larger risk aversion strengthens optimal pollution control. Second, larger resistance to substitution — more emphasis on intergenerational equity — rotates the optimal control path toward less control in the current decade and more control in the future. This compares with an increase in the utility discount rate, but the effects differ from a sustainability point of view. Third and finally, low resistance to substitution - with high risk aversion leads to the largest carbon tax levels. Constraining - Û leads to underestimating the sensitivity of results to the shape of the utility function..

(85) 22. HA DUONG AND TREICH. APPENDIX:. DE:G< 7 ?U:< 7 9;:< 7 =0 :< 7 7 Ü :G< 7 o :< 7 p :<7 :< sI:G< 7 AB:< 7 xz:< 7 ^:< 7. THE ULTRAL MODEL Variables. Energy cost multiplier (Control variable) Production (PIB per capita per year in kUS$ 90) Capital stock (kUS$ 90 per capita) Energy input (toe per capita per year) Consumption (kUS$ 90 per capita per year) Adjustment costs factor (% of investment) Begin-of-period cuMulative global emissions (GtC) 021 Begin-of-period mixing ratio (ppmv) Equilibrium temperature increase ( Ý C) Climate damage (fraction of ?:G< 7 ) Utility (k dol. 90 per capita) Certainty equiValent of next generation’s utility (kUS$ 90 p.c.). Equations ULTRAL maximizes x hi<

(86) under constraints A.1-A.11, the stochastic dynamic programming constraint D j'< 7 D hi< 7 for all 3Ek d'f and the final stationarity condition D :G< D :G< « .. ?U:< 7 J ?

(87) 0 7 :< & 7 J ? :< ?

(88) 9 :< 7Gy

(89) 7Gy

(90) Ü 7Gy :G<

(91) o. @ 7 At:< 7 9;:< 7 J 9

(92) Þ =V:G< 7. ß C 7 Ü 7 ? :G< 7. F 7 D :G< 7 = :< 7J =

(93). ß ¹ 9 :< 7 ! º²C 7 ? :G< 7 & 7. áàEâãg ä DE:G< 7 J DE:< 7 g

(94) . J=

(95) L. (A.1) (A.2) (A.3) (A.4). å( Í J æ ä. o 7 :G< 7Gy

(96) . :< !Hç ° 7 =>:< 7R87 p H o :< 7 q 7r ! q 7

(97) :G< 7 p s :< 7 è³~´² :< 7 ³NéŃ J ³Né´ if 3lì dgu í 7 dgu. ë. ê 7 s V:< A :< í 7 if 3lî d u and 6. J v s :< if 3lî and 6 ! v 7J 0

(98) 7

(99) x :< 7 :< ! :< 7 (+*#( , xz:< 7Gy

(100) 87Gy

(101) ð U:G< 7R87) ê v 87Gy

(102) xzhi< 7Gy

(103)

(104) ! v 87Gy

(105) xzjg< 7Gy 0 :G< ¹ 9 :< ! º^C ? :G< J 9 :G< . (A.5) (A.6) (A.7) (A.8). 4. (A.9). 5. (

(106)

(107) UNñ (Q*. ï d f if 3we if 3 ed f if 3 em. (A.10). and 3w e ï m. (A.11).

(108) RECURSIVE INTERGENERATIONAL UTILITY. 23. Parameters (A1 scenario). ¶ 8 7. Elasticity of capital Elasticity of energy World population Technical progress coefficient Saving ratio Baseline increase in the energy input cost Production per capita in 1990 without damages Energy per capita in 1990 without reduction Initial capital Inter-period depreciation of capital Inter-period accumulation of investment Maximum X annual variation without adj. costs. 0.36 0.04 5.26, 7.49, 8.70, 7.72 Giga 1.00, 1.81, 3.69, 5.52 0.25 1.00, 1.28, 2.05, 2.70 3.97 k$90 1.07 toe 11.92 k$90 0.215 16.14 0.02. . Inter-period utility discount factor Resistance to intertemporal substitution Relative risk aversion. 0.74 1.3 2.0. ç ° 7 qò7r q 7

(109) d u dgf. Duration of a period Carbon content of energy Intercept of the carbon-cycle relation Slope of the carbon-cycle relation Date of impact: productivity jump Event Date of policy Revision Probability for a negative impact Magnitude of damage for 1 Ý C warming. 30 years 1.10, 1.03, 0.68, 0.39 tC/toe 354, 371, 390, 297 ppmv 0, 0.154, 0.129, 0.278 ppmv/toe 4 4 0.05 0.1. T. @7 C7 F 7 ?

(110) =

(111) 9

(112) ¹ º æ . -. ví.

(113) 24. HA DUONG AND TREICH. Scenarios Data from IPCC preliminary 1998 scenarios used for calibration. Year 2080 values are linearly interpolated except for emissions. Energy Intensity in B1 taken from A1.. _&Ùó. 1990. 2020. 2050. 2080. 2100. A1. Population GNP per capita Energy Intensity. 1990k$ MJ/$. 5.3 4.0 11.3. 7.5 7.5 8.8. 8.7 20.8 5.5. 7.7 50.5 4.2. 7.1 74.9 3.3. A2. Population GNP per capita Energy Intensity. 1990k$ MJ/$. 5.3 4.0 12.8. 8.2 5.0 12.4. 11.3 7.3 10.0. 12.4 8.9 7.4. 13.1 9.9 5.7. B1. Population GNP per capita Energy Intensity. 1990k$ MJ/$. 5.3 4.0 11.3. 7.8 6.2 8.8. 8.9 12.8 5.5. 7.9 31.4 4.2. 7.2 46.7 3.3. B2. Population GNP per capita Energy Intensity. 1990k$ MJ/$. 5.3 4.0 12.9. 7.7 6.6 8.5. 9.4 11.7 6.0. 10.0 18.5 4.8. 10.4 22.6 4.0. _&Ù ó _&Ù ó _&Ù ó.

(114) RECURSIVE INTERGENERATIONAL UTILITY. 25. REFERENCES 1. Kenneth J. Arrow. Discounting, Morality and Gaming, chapter 2. In Portney and Weyant [30], 1999. 2. Kenneth J. Arrow, W. R. Cline, K.-G. Maler, M. Munasinghe, R. Squitieri, and J. E. Stiglitz. Equité entre générations, actualisation et efficacité e´ conomique, chapter 4, pages 121–143. In Bruce et al. [5], 1996. English edition published by Cambridge University Press. 3. Geir B. Asheim and Kjell Arne Brekke. Sustainability when capital management has stochastic consequences. Memorandum 9, Departement of Economics, University of Oslo, April 1997. ISSN=0801-1117. 4. R. B. Barsky, F. T. Juster, M. Kimball, and M. D. Shapiro. Preferences parameters and behavioral heterogeneity: An experimental approach in the health and retirement study. Quarterly Journal of Economics, pages 537–579, 1997. 5. James P. Bruce, Hoesung Lee, and Erik F. Haites, editors. Le Changement Climatique. Dimensions e´ conomiques et sociales. Contribution du Groupe de Travail III au Deuxième Rapport d’évaluation du du Groupe d’Experts Intergouvernemental sur l’évolution du climat. Dossiers et débats pour le développement durable (4D), 1996. English edition published by Cambridge University Press. 6. William R. Cline. The Economics of global Warming. Institute for International Economics, Washington, DC, June 1992. 7. William R. Cline. Discounting for the Very Long Run, chapter 13. In Portney and Weyant [30], 1999. 8. I. G. Enting, T. M. L. Wigley, and M. Heimann. Future emissions and concentrations of carbon dioxide: Key Ocean/Atmosphere/Land analyses. Technical Report Technical Paper 31, CSIRO Division of Atmospheric Research, 1994. 9. Anne Epaulard and Aude Pommeret. Does uncertainty lead to a more conservative use of a ´ non renewable resource? A recursive utility approach. In Journées de l’AFSE sur \Economie de l’Environnement et des Ressources Naturelles", 11-12 mai 1998. 10. L. G. Epstein and S. E. Zin. Substitution, risk-aversion and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica, 57:937–969, 1989. 11. L. G. Epstein and S. E. Zin. Substitution, risk-aversion and the temporal behavior of consumption and asset returns: An empirical analysis. Journal of Political Economy, 99:263–286, 1991. ´ 12. Christian Gollier. Discounting an uncertain future. In Journées de l’AFSE sur \Economie de l’Environnement et des Ressources Naturelles", 11-12 mai 1998. 13. Christian Gollier, Bruno Jullien, and Nicolas Treich. Scientific progress and irreversibility: an economic interpretation of the “precautionary principle”. Journal of Public Economics, 1999. forthcoming. 14. R. E. Hall. Intertemporal substitution in consumption. Journal of Political Economy, 96:338–357, 1988. 15. K. C. Knapp and L. J. Olson. Dynamic resource management: Intertemporal substitution and risk aversion. American Journal of Agricultural Economics, 78:1004–1114, 1996. 16. Charles D. Kolstad. Learning and stock effects in environmental regulation: The case of greenhouse gas emisions. Journal of Environmental Economics and Management, 31:1–18, 1996. 17. Tjalling C. Koopmans. Stationary ordinal utility and impatience. Econometrica, 28(2):287–309, 1960. 18. D. Kreps and E. Porteus. Temporal resolution of uncertainty and dynamic choice theory. Econometrica, 46:185–200, 1978. 19. Robert C. Lind. Intergenerational equity, discounting, and the role of cost-benefit analysis in evaluating global climate policy. Energy Policy, 23:379–389, 1995..

(115) 26. HA DUONG AND TREICH. 20. Alan S. Manne and Richard Richels. Buying Greenhouse Insurance: The Economic Cost of Emissions Limits. MIT Press, 1992.. ôzõ Æ. 21. R. Mehra and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics, 15:145–161, 1985. 22. Robert Mendelsohn and Michael E. Schlesinger. Climate-response functions. Ambio, 28(4):362– 366, jun 1999. 23. Ha Duong Minh, Michael J. Grubb, and Jean-Charles Hourcade. Influence of socioeconomic inertia and uncertainty on optimal -emission abatement. Nature, 390:270–274, 1997. Also available electronically as .pdf from Nature website.. ôzõ Æ. 24. William D. Nordhaus. Managing the Global Commons, The Economics of Climate Change. The MIT Press, Cambridge, Massachusetts, 1994. 25. William D. Nordhaus. Notes on scenarios for uncertainty subgroup. Working paper, Yale University, New Have, CT, USA, jun 1995. 26. William D. Nordhaus. Discounting and Public Policies That Affect the Distant Future, chapter 15. In Portney and Weyant [30], 1999. 27. M. Normandin and P. Saint-Amour. Substitution, risk aversion, taste shocks and equity premia. Journal of Applied Econometrics, pages 265–281, 1998. 28. M. Ogaki and M. R. Reinhart. Measuring intertemporal substitution: The role of durable goods. Journal of Political Economy, pages 1078–1098, 1998. 29. J. C. V. Pezzey. Sustainability constraints versus ‘optimality’ versus intertemporal concern, and axioms versus data. Land Economics, 73:448–466, 1997. 30. Paul R. Portney and John P. Weyant, editors. Discounting and Intergenerational Equity. Resources for the Future, Washington, DC, 1999.. ö. 31. L. Selden. A new representation of preferences over ‘certain uncertain’ pairs: The ‘ordinal certainty equivalent hypothesis’. Econometrica, 16:1045,1060, 1978. 32. R. M. Solow. Sustainability: An economist’s perspective. In R. Dorfman and N. S. Dorfman, editors, Economics of the Environment. Norton, New York, 1993. 33. Robert M. Solow. Intergenerational equity and exhaustible resources. Review of Economic Studies, 41:29–45, 1974. 34. P. Weil. Nonexpected utility in macroeconomics. Quarterly Journal of Economics, pages 29–42, 1990. 35. M. Weitzman. Why the far-distant future should be discounted at its lowest possible rate ? Journal of Environmental Economics and Management, 36:210–208, 1998..

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